Quality ControlQuality Control

Dr. Everette S. Gardner, Jr.

Quality 2

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Engineering characteristics

Customer requirements

Importance to customer

54321

Easy to closeStays open on a hillEasy to open

Doesn’t leak in rainNo road noise

Importance weighting

75332

10 6 6 9

Source: Based on John R. Hauser and Don Clausing, “The House of Quality,” Harvard Business Review, May-June 1988.

2 3

x

xx x

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Competitive evaluationxAB(5 is best)1 2 3 4 5

= Us= Comp. A= Comp. B

Target values

Technical evaluation (5 is best)

Correlation:Strong positivePositiveNegativeStrong negative

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Relationships:Strong = 9

Medium = 3Small = 1

Quality 3

Taguchi analysisLoss functionL(x) = k(x-T)2

wherex = any individual value of the quality characteristicT = target quality valuek = constant = L(x) / (x-T)2

Average or expected loss, variance knownE[L(x)] = k(σ2 + D2)where

σ2 = Variance of quality characteristic D2 = ( x – T)2

Note: x is the mean quality characteristic. D2 is zero if the mean equals the target.

Quality 4

Taguchi analysis (cont.)

Average or expected loss, variance unkownE[L(x)] = k[Σ ( x – T)2 / n]

When smaller is better (e.g., percent of impurities)L(x) = kx2

When larger is better (e.g., product life)L(x) = k (1/x2)

Quality 5

Introduction to quality control charts

Definitions• Variables Measurements on a continuous scale, such as length or

weight• Attributes Integer counts of quality characteristics, such as nbr.

good or bad• Defect A single non-conforming quality characteristic, such as a

blemish• Defective A physical unit that contains one or more defects

Types of control charts

Data monitored Chart name Sample size

• Mean, range of sample variables MR-CHART 2 to 5 units• Individual variables I-CHART 1 unit• % of defective units in a sample P-CHART at least 100

units• Number of defects per unit C/U-CHART 1 or more units

Quality 6

Sample mean value

Sample number

99.74%

0.13%

0.13%

Upper control limit

Lower control limit

Process mean

Normaltolerance

ofprocess

0 1 2 3 4 5 6 7 8

Quality 7

Reference guide to control factorsn A A2 D3 D4 d2 d3

2 2.121 1.880 0 3.267 1.128 0.8533 1.732 1.023 0 2.574 1.693 0.8884 1.500 0.729 0 2.282 2.059 0.8805 1.342 0.577 0 2.114 2.316 0.864

• Control factors are used to convert the mean of sample ranges ( R ) to:

(1) standard deviation estimates for individual observations, and(2) standard error estimates for means and ranges of samples

For example, an estimate of the population standard deviation of individual observations (σx) is:

σx = R / d2

Quality 8

Reference guide to control factors (cont.)

• Note that control factors depend on the sample size n.

• Relationships amongst control factors:A2 = 3 / (d2 x n1/2)

D4 = 1 + 3 x d3/d2

D3 = 1 – 3 x d3/d2, unless the result is negative, then D3 = 0

A = 3 / n1/2

D2 = d2 + 3d3

D1 = d2 – 3d3, unless the result is negative, then D1 = 0

Quality 9

Process capability analysis

1. Compute the mean of sample means ( X ).

2. Compute the mean of sample ranges ( R ).

3. Estimate the population standard deviation (σx):

σx = R / d2

4. Estimate the natural tolerance of the process:Natural tolerance = 6σx

5. Determine the specification limits:USL = Upper specification limitLSL = Lower specification limit

Quality 10

Process capability analysis (cont.)

6. Compute capability indices:Process capability potential

Cp = (USL – LSL) / 6σx

Upper capability indexCpU = (USL – X ) / 3σx

Lower capability indexCpL = ( X – LSL) / 3σx

Process capability indexCpk = Minimum (CpU, CpL)

Quality 11

Mean-Range control chartMR-CHART

1. Compute the mean of sample means ( X ).

2. Compute the mean of sample ranges ( R ).

3. Set 3-std.-dev. control limits for the sample means:UCL = X + A2R

LCL = X – A2R

4. Set 3-std.-dev. control limits for the sample ranges:UCL = D4R

LCL = D3R

Quality 12

Control chart for percentage defective in a sample — P-CHART

1. Compute the mean percentage defective ( P ) for all samples:P = Total nbr. of units defective / Total nbr. of units

sampled

2. Compute an individual standard error (SP ) for each sample:

SP = [( P (1-P ))/n]1/2

Note: n is the sample size, not the total units sampled.If n is constant, each sample has the same standard

error.

3. Set 3-std.-dev. control limits:UCL = P + 3SP

LCL = P – 3SP

Quality 13

Control chart for individual observations — I-CHART

1. Compute the mean observation value ( X )X = Sum of observation values / Nwhere N is the number of observations

2. Compute moving range absolute values, starting at obs. nbr. 2:

Moving range for obs. 2 = obs. 2 – obs. 1Moving range for obs. 3 = obs. 3 – obs. 2…Moving range for obs. N = obs. N – obs. N – 1

3. Compute the mean of the moving ranges ( R ):R = Sum of the moving ranges / N – 1

Quality 14

Control chart for individual observations — I-CHART (cont.)

4. Estimate the population standard deviation (σX):

σX = R / d2

Note: Sample size is always 2, so d2 = 1.128.

5. Set 3-std.-dev. control limits:UCL = X + 3σX

LCL = X – 3σX

Quality 15

Control chart for number of defects per unit — C/U-CHART1. Compute the mean nbr. of defects per unit ( C ) for all samples:

C = Total nbr. of defects observed / Total nbr. of units sampled

2. Compute an individual standard error for each sample:SC = ( C / n)1/2

Note: n is the sample size, not the total units sampled.If n is constant, each sample has the same standard error.

3. Set 3-std.-dev. control limits:UCL = C + 3SC

LCL = C – 3SC

Notes:● If the sample size is constant, the chart is a C-CHART.● If the sample size varies, the chart is a U-CHART.● Computations are the same in either case.

Quality 16

Quick reference to quality formulas

• Control factorsn A A2 D3 D4 d2 d3

2 2.121 1.880 0 3.267 1.128 0.8533 1.732 1.023 0 2.574 1.693 0.8884 1.500 0.729 0 2.282 2.059 0.8805 1.342 0.577 0 2.114 2.316 0.864

• Process capability analysis σx = R / d2

Cp = (USL – LSL) / 6σx CpU = (USL – X ) / 3σx

CpL = ( X – LSL) / 3σx Cpk = Minimum (CpU, CpL)

Quality 17

Quick reference to quality formulas (cont.)

• Means and rangesUCL = X + A2R UCL = D4RLCL = X – A2R LCL = D3R

• Percentage defective in a sample SP = [( P (1-P ))/n]1/2 UCL = P + 3SP

LCL = P – 3SP

• Individual quality observations σx = R / d2 UCL = X + 3σX

LCL = X – 3σX

• Number of defects per unitSC = ( C / n)1/2 UCL = C + 3SC

LCL = C – 3SC

Quality 18

Multiplicative seasonality

The seasonal index is the expected ratio of actual data to the average for the year.

Actual data / Index = Seasonally adjusted data

Seasonally adjusted data x Index = Actual data

Quality 19

Multiplicative seasonal adjustment1. Compute moving average based on length of

seasonality (4 quarters or 12 months).

2. Divide actual data by corresponding moving average.

3. Average ratios to eliminate randomness.

4. Compute normalization factor to adjust mean ratios so they sum to 4 (quarterly data) or 12 (monthly data).

5. Multiply mean ratios by normalization factor to get final seasonal indexes.

6. Deseasonalize data by dividing by the seasonal index.

7. Forecast deseasonalized data.

8. Seasonalize forecasts from step 7 to get final forecasts.

Quality 20

Additive seasonality

The seasonal index is the expected difference between actual data and the average for the year.

Actual data - Index = Seasonally adjusted data

Seasonally adjusted data + Index = Actual data

Quality 21

Additive seasonal adjustment

1. Compute moving average based on length of seasonality (4 quarters or 12 months).

2. Compute differences: Actual data - moving average.

3. Average differences to eliminate randomness.

4. Compute normalization factor to adjust mean differences so they sum to zero.

5. Compute final indexes: Mean difference – normalization factor.

6. Deseasonalize data: Actual data – seasonal index.

7. Forecast deseasonalized data.

8. Seasonalize forecasts from step 7 to get final forecasts.

Quality 22

How to start up a control chart system1. Identify quality characteristics.

2. Choose a quality indicator.

3. Choose the type of chart.

4. Decide when to sample.

5. Choose a sample size.

6. Collect representative data.

7. If data are seasonal, perform seasonal adjustment.

8. Graph the data and adjust for outliers.

Quality 23

How to start up a control chart system (cont.)9. Compute control limits

10. Investigate and adjust special-cause variation.

11. Divide data into two samples and test stability of limits.

12. If data are variables, perform a process capability study:a. Estimate the population standard deviation.b. Estimate natural tolerance.c. Compute process capability indices.d. Check individual observations against

specifications.

13. Return to step 1.